Hybrids running on battery are about as efficient as an EV. When you enter the highway, they turn into ordinary ICE engines. If you use your prius exclusively for traffic in your neighborhood, you only use petrol to charge the battery, which is efficient and about as good as it gets with a hybrid. Unless you can plug it in of course. It won't use any petrol at all in that case.

Modern cars with good high-compression engines have efficiencies over 40%.

A fuel cell with hydrocarbons could reach efficiencies of 60% or more.

So no lithium battery can reach volumic energies or specific energies comparable to what can be achieved with hydrocarbons.

The reason to use lithium rechargeable batteries is to obtain a better total efficiency of using energy, not the hope that it is possible to match the densities achievable with energy stored in hydrocarbons.

Among lithium rechargeable batteries, the lithium-air batteries should achieve the best energy per mass, perhaps also per volume.

Usually the weak point of metal-air batteries is the power per mass or the power per volume, because the reaction with air is slow, therefore the electrical current density in the electrodes is low, so to obtain a given amount of power requires great areas for the electrodes.

To do that, you need to expend roughly the same amount of energy that is needed to first liquify and then solidify the oxygen.

In fancy chemistry-speak it's called "entropic loss". You do gain some of that energy back when the battery is charged, as oxygen goes from a well-ordered solid state into the gaseous state. But it's not 100%.

That claim is based on dividing the stored energy by the mass of lithium, which is incorrect.

The product of the reaction, i.e. lithium oxide, is stored in the battery, so a lithium-air battery can never be lighter than the lithium oxide.

Because the mass of lithium oxide is what counts, the energy per mass of pure lithium, which is indeed not much less than for gasoline, must be divided by a factor that varies between 2.14 and 5.57, depending on the construction of the lithium-air battery.

The best value of 2.14 is when the discharged battery contains only Li2O. The worst value of 5.57 is when the discharged battery contains only lithium superoxide, LiO2.

In the parent article, they claim that their discharged battery contains mostly Li2O, with only small quantities of peroxide Li2O2 and superoxide LiO2, but the exact amounts of peroxide and superoxide have not been measured.

So when computing correctly the energy per mass ratio, for lithium-air batteries it is limited to a value less than half of that for hydrocarbons. In practice batteries need a lot of materials besides the active reactants, so the achievable energy per mass ratio will be several times lower.

The advantage of hydrocarbons, regardless whether they are used in living cells, thermal engines or fuel cells, is that their reaction products are eliminated into the atmosphere, so their mass does not matter. The energy per mass for carbon atoms in hydrocarbons and for lithium atoms in lithium metal is approximately the same, but with lithium it is impossible to neglect the mass of the oxidant, like with carbon, because the reaction products cannot be dumped outside.

So for any battery except for fuel cells, what counts is the sum of the masses of the reactants, e.g. lithium + oxygen in the best case, or e.g. zinc + manganese in the cheap non-rechargeable batteries. It is wrong to compute the minimum mass of a battery by using only the mass of one of the reactants, like in the parent article, instead of both masses.

For any kind of battery, there will be a power threshold over which a fuel tank + a turbo-generator will be smaller and lighter.

So a useful comparison would determine those power thresholds.